\documentclass[fleqn, 12pt]{article}
\pagestyle{empty}
\setlength{\topmargin}{-.5in}
\setlength{\oddsidemargin}{-.3in}
\setlength{\textheight}{9in}
\setlength{\textwidth}{7.2in}


\usepackage{amssymb}
\usepackage{amsmath}


\newcommand{\adj}{\mathrm{Adj}}
\newcommand{\mbf}{\mathbf}
\newcommand{\R}{\mathbb{R}}
\newcommand{\ra}{\rightarrow}
\newcommand{\proj}{\mathrm{proj}}
\newcommand{\dotprod}{\bullet}

\newcommand{\sol}{{\tiny\textbf{Solution}}}


\begin{document}

\begin{center}
{\Large Warm-Up Problems

January 31, 2003}
\end{center}


\begin{enumerate}

\item  Let $f(x)=x^3$.  Compute the following.  Think about what is happening geometrically as well.
	\begin{enumerate}
	\item  $\int_{-1}^{1} f(x) \; dx$
		\\ \sol: $0$
	\item  $\int_{-1}^{1} |f(x)| \; dx$
		\\ \sol: $\frac{1}{2}$
	\item  $\int_{-3}^{1} f(x) \; dx$
		\\ \sol: $-20$
	\item  $\int_{-1}^{3} |f(x)| \; dx$
		\\ \sol: $\frac{41}{2}$
	\end{enumerate}

\item  Compute the following integrals:
	\begin{enumerate}
	\item  $\int_0^{\pi/2} \cos x \sqrt{\sin x} \; dx $
		\\ \sol: $\frac{2}{3}$
	\item  $\int_1^4 \frac{((1+\sqrt{x})^4}{\sqrt{x}} \; dx$'
		\\ \sol: $\frac{422}{5}$
	\item  $\int_0^{\pi/6} \sin 2x \cos^3 2x \; dx$
		\\ \sol: $\frac{15}{128}$
	\item \label{tan} 
		$\int \tan x \; dx$
		\\ \sol: $-\ln|\cos x| +C$
	\item \label{sini} 
		$\int \frac{x}{\sqrt{1-9x^4}} \;dx$
		\\ \sol: $\frac{1}{9}\sin^{-1}(3x^2)+C$
	\end{enumerate}

\end{enumerate}

\bigskip
\bigskip

Hints: for number~\ref{tan}, try $u=\cos x$.
	For number~\ref{sini}, try $u=3x^2$.
	Why do these work?


\end{document}